Effect of transverse acoustic flow on the input impedance of rapidly flaring horns.
Cornelis J. Nederveen

For most wind instruments, the waves in the inside of the pipe are approximately plane. The field may be considered one-dimensional; the input impedance can be calculated by describing the pipe as a succession of short cones (transmission line method). This is less accurate for fast changes in the bore, for example in a rapidly flaring horn, where the kinetic energy of the transverse acoustic flow is no longer small compared with that of the longitudinal flow, causing a local increase in inertia, resulting in an effective increase in length when the horn is located near a pressure node. For higher frequencies, where cross-dimensions become comparable to the wavelength, resonances can occur in the cross-direction. To investigate this, the pipe radiating in open space was modelled with a finite difference method. Because of limits in computer capacity the outer domain is bounded. To avoid reflections its boundaries must be fully absorbing just as the walls of an anechoic chamber. The well-known Sommerfeld radiation condition is only useful at low frequencies, reflections occur at higher frequencies. Much more effective is Berenger’s PML (perfectly matched layer); it is applied here. Presented are results for various horns, they are compared with earlier published investigations on flanges. In all cases the inertance exhibits a maximum when the largest cross-dimension (outer diameter of flange or diameter of the horn end) becomes comparable to half a wavelength. This effect changes the position of higher modes in the pipe, influencing the conditions for mode locking, important for ease of playing, dynamic range and sound quality.